A383530 Number of non Wilf and non conjugate Wilf integer partitions of n.

0, 0, 0, 1, 0, 0, 3, 2, 5, 12, 14, 19, 35, 38, 55, 83, 107, 137, 209, 252, 359, 462, 612, 757, 1032, 1266, 1649, 2050, 2617, 3210, 4111, 4980, 6262, 7659, 9479, 11484, 14224, 17132, 20962, 25259, 30693, 36744, 44517, 53043, 63850, 75955, 90943, 107721, 128485

Offset: 0

Author

Gus Wiseman, May 14 2025

Keywords

• nonn

Comments

An integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its nonzero 0-appended differences are all different (ranked by A383512).

Examples

			The a(0) = 0 through a(9) = 12 partitions:
  .  .  .  (21)  .  .  (42)    (421)   (431)    (63)
                       (321)   (3211)  (521)    (432)
                       (2211)          (3221)   (531)
                                       (4211)   (621)
                                       (32111)  (3321)
                                                (4221)
                                                (4311)
                                                (5211)
                                                (32211)
                                                (42111)
                                                (222111)
                                                (321111)
		

Links

• Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs

Negating both sides gives A383507, ranks A383532.

These partitions are ranked by A383531.

A048767 is the Look-and-Say transform, union A351294, complement A351295.

A098859 counts Wilf partitions, ranks A130091, conjugate A383512.

A239455 counts Look-and-Say partitions, complement A351293.

A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.

A381431 is the section-sum transform, union A381432, complement A381433.

A383534 gives 0-prepended differences by rank, see A325351.

A383709 counts Wilf partitions with distinct 0-appended differences, ranks A383712.

Cf. A033461, A047966, A111133, A320348, A325324, A325325, A325349, A325367, A325368, A325388, A383506.

Programs

Mathematica

conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]], {k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n], !UnsameQ@@Length/@Split[#]&&!UnsameQ@@Length/@Split[conj[#]]&]], {n,0,30}]

Formula

These partitions have Heinz numbers A130092 /\ A383513.